\begin{proof} 1. $\emptyset,\Omega \in \mathcal{J}$ 2. Closed under intersection since each event is either disjoint (i.e. its made up of discrete permutation of events) or they’re the same meaning the intersection is the same. 3. Complement=finite union: 1. Use idea that $(0,0)^{c}=(0,1)\cup(1,1)\cup(1,0)$ and then apply this at level $n$. \end{proof}

\begin{proof} 1: Use Extension Theorem, need to show $\mathbb{P}$ is countably additive on $\mathcal{J}$. Clearly, $\mathbb{P}$ is finitely additive: $$\begin{align*} \mathbb{P}(A_{a_{1},a_{2}}\cup A_{a_{1},a_{2}^{c}})=\mathbb{P}(A_{a_{1}})=\frac{1}{2}\\ \mathbb{P}(A_{a_{1},a_{2}})+\mathbb{P}(A_{a_{1},a_{2}^{c}})=\frac{1}{2^{2}}+\frac{1}{2^{2}}=\frac{1}{2} \end{align*}$$ Countably requires Compactness argument (see lemma 2.62 in book) 2: Compare $\Omega$ to $(0,1]$ via binary expansion. \end{proof} >[!lemma] >Every $x \in(0,1]$ has a unique binary expansion $$x=\sum_{j>1}\frac{r_{j}}{2^{j}}$$such that $r_{j}\in \{ 0,1 \}$ and $r_{j}=1$ i.o.. >